Question 9
Given a nonnegative real number $x,$ let $\langle x\rangle$ denote the fractional part of $x;$ that is, $\langle x\rangle = x - \lfloor x\rfloor,$ where $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x.$ Suppose that $a$ is positive, $\langle a^{-1}\rangle = \langle a^2\rangle,$ and $2 \lt a^2 \lt 3.$ Find the value of $a^{12} - 144a^{-1}.$